Book:George B. Thomas, Jr./Calculus and Analytic Geometry/Fourth Edition

George B. Thomas, Jr.: Calculus and Analytic Geometry (4th Edition)

Published $\text {1968}$, Addison Wesley

Subject Matter

• Calculus
• Analytic Geometry

Contents

Preface to the Fourth Edition

$1 \quad$ The rate of change of a function

$1.1 \quad$ Introduction

$1.2 \quad$ Coordinates

$1.3 \quad$ Increments

$1.4 \quad$ Slope of a straight line

$1.5 \quad$ Equations of a straight line

$1.6 \quad$ Functions and graphs

$1.7 \quad$ Ways of combining functions

$1.8 \quad$ Behavior of functions

$1.9 \quad$ Slope of a curve

$1.10 \quad$ Derivative of a function

$1.11 \quad$ Velocity and rates

$2 \quad$ Limits

$2.1 \quad$ Definition of the limit of a function

$2.2 \quad$ Theorems about limits

$2.3 \quad$ More theorems about limits

$2.4 \quad$ Infinity

$2.5 \quad$ Limits applied to areas

$3 \quad$ Derivatives of Algebraic Functions

$3.1 \quad$ Polynomial functions and their derivatives

$3.2 \quad$ Rational functions and their derivatives

$3.3 \quad$ Inverse functions and their derivatives

$3.4 \quad$ The increment of a function

$3.5 \quad$ Composite functions

$3.6 \quad$ Derivatives of composite functions: the chain rule

$3.7 \quad$ The differentials $dx$ and $dy$

$3.8 \quad$ Formulas for differentiation

$3.9 \quad$ Continuity

$4 \quad$ Applications

$4.1 \quad$ Increasing or decreasing functions: the sign of $dy / dx$

$4.2 \quad$ Related rates

$4.3 \quad$ Significance of the sign

$4.4 \quad$ Curve plotting

$4.5 \quad$ Maxima and minima: theory

$4.6 \quad$ Maxima and minima: problems

$4.7 \quad$ Rolle's Theorem

$4.8 \quad$ The Mean Value Theorem

$4.9 \quad$ Extension of the Mean Value Theorem

$5 \quad$ Integration

$5.1 \quad$ Introduction

$5.2 \quad$ The indefinite integral

$5.3 \quad$ Applications of indefinite integration

$5.4 \quad$ Brief review of trigonometry

$5.5 \quad$ Differentiation and integration of sines and cosines

$5.6 \quad$ Area under a curve

$5.7 \quad$ Computation of areas as limits

$5.8 \quad$ Areas by calculus

$5.9 \quad$ The definite integral and the Fundamental Theorem of Integral Calculus

$5.10 \quad$ The trapezoidal rule for approximating an integral

$5.11 \quad$ Some comments on notation

$5.12 \quad$ Summary

$6 \quad$ Applications of the definite integral

$6.1 \quad$ Introduction

$6.2 \quad$ Area between two curves

$6.3 \quad$ Distance

$6.4 \quad$ Volumes

$6.5 \quad$ Approximations

$6.6 \quad$ Length of a plane curve

$6.7 \quad$ Area of a surface of revolution

$6.8 \quad$ Average value of a function

$6.9 \quad$ Moments and center of mass

$6.10 \quad$ The centroid

$6.11 \quad$ The theorems of Pappus

$6.12 \quad$ Hydrostatic pressure

$6.13 \quad$ Work

$7 \quad$ Transcendental functions

$7.1 \quad$ The trigonometric functions

$7.2 \quad$ The inverse trigonometric functions

$7.3 \quad$ Derivatives of the inverse trigonometric functions

$7.4 \quad$ The natural logarithm

$7.5 \quad$ The derivative of $\ln x$

$7.6 \quad$ Properties of natural logarithms

$7.7 \quad$ Graph of $y = \ln x$

$7.8 \quad$ The exponential function

$7.9 \quad$ The functions $a^u$ and $\log_a u$

$7.10 \quad$ Differential equations

$8 \quad$ Hyperbolic functions

$8.1 \quad$ Introduction

$8.2 \quad$ Definitions and identities

$8.3 \quad$ Derivatives and integrals

$8.4 \quad$ Geometric meaning of the hyperbolic radian

$8.5 \quad$ The inverse hyperbolic functions

$8.6 \quad$ The hanging cable

$9 \quad$ Methods of integration

$9.1 \quad$ Basic formulas

$9.2 \quad$ Powers of trigonometric functions

$9.3 \quad$ Even powers of sines and cosines

$9.4 \quad$ Integrals with terms $\sqrt {a^2 - u^2}$, $\sqrt {a^2 + u^2}$, $\sqrt {u^2 - a^2}$, $a^2 + u^2$, $a^2 - u^2$

$9.5 \quad$ Integrals with $a x^2 + b x + c$

$9.6 \quad$ Integration by the method of partial fractions

$9.7 \quad$ Integration by parts

$9.8 \quad$ Integration of rational functions of $\sin x$ and $\cos x$, and other trigonometric integrals

$9.9 \quad$ Further substitutions

$9.10 \quad$ Improper integrals

$9.11 \quad$ Numerical methods for approximating definite integrals

$10 \quad$ Plane analytic geometry

$10.1 \quad$ Curves and equations

$10.2 \quad$ Tangents and normals

$10.3 \quad$ Newton's method for approximating roots of equations

$10.4 \quad$ Distance between two points: equations of loci

$10.5 \quad$ The circle

$10.6 \quad$ The parabola

$10.7 \quad$ The ellipse

$10.8 \quad$ The hyperbola

$10.9 \quad$ Second-degree curves

$10.10 \quad$ Invariants and the discriminant

$10.11 \quad$ Sections of a cone

$11 \quad$ Polar coordinates

$11.1 \quad$ The polar coordinate system

$11.2 \quad$ Graphs of polar equations

$11.3 \quad$ Polar equations of the conic sections and other curves

$11.4 \quad$ The angle $\bspsi$ between the radius vector and the tangent line

$11.5 \quad$ Plane areas in polar coordinates

$12 \quad$ Vectors and parametric equations

$12.1 \quad$ Parametric equations in kinematics

$12.2 \quad$ Parametric equations in analytic geometry

$12.3 \quad$ Vector components and the unit vectors $\mathbf i$ and $\mathbf j$

$12.4 \quad$ Space coordinates

$12.5 \quad$ Vectors in space

$12.6 \quad$ The scalar product of two vectors

$12.7 \quad$ The vector product of two vectors

$12.8 \quad$ Equations of lines and planes

$12.9 \quad$ Products of three or more vectors

$12.10 \quad$ Loci in space: cylinders

$12.11 \quad$ Quadric surfaces

$13 \quad$ Linear algebra: vectors in $n$-space

$13.1 \quad$ Vectors in Euclidean $n$-space

$13.2 \quad$ Matrices and simultaneous linear equations: notation

$13.3 \quad$ Matrices and simultaneous linear equations: computational techniques

$13.4 \quad$ Linear independence and linear dependence of vectors

$13.5 \quad$ Matrices and linear transformations

$14 \quad$ Vector functions and their derivatives

$14.1 \quad$ Introduction

$14.2 \quad$ Velocity and acceleration

$14.3 \quad$ Tangential vectors

$14.4 \quad$ Curvature and normal vectors

$14.5 \quad$ Differentiation of products of vectors

$14.6 \quad$ Polar and cylindrical coordinates

$15 \quad$ Partial differentiation

$15.1 \quad$ Functions of two or more variables

$15.2 \quad$ The direction derivative: special cases

$15.3 \quad$ Tangent plane and normal line

$15.4 \quad$ Approximate value of $\Delta w$

$15.5 \quad$ The directional derivative: general case

$15.6 \quad$ The gradient

$15.7 \quad$ The chain rule for partial derivatives

$15.8 \quad$ The total differential

$15.9 \quad$ Maxima and minima of functions of two independent variables

$15.10 \quad$ The method of least squares

$15.11 \quad$ Maxima and minima of functions of several independent variables

$15.12 \quad$ Higher-order derivatives

$15.13 \quad$ Exact differentials

$15.14 \quad$ Derivatives of integrals

$16 \quad$ Multiple integrals

$16.1 \quad$ Double integrals

$16.2 \quad$ Area by double integration

$16.3 \quad$ Physical applications

$16.4 \quad$ Polar coordinates

$16.5 \quad$ Triple integrals: volume

$16.6 \quad$ Cylindrical coordinates

$16.7 \quad$ Physical applications of triple integration

$16.8 \quad$ Spherical coordinates

$16.9 \quad$ Surface area

$17 \quad$ Vector analysis

$17.1 \quad$ Introduction: vector fields

$17.2 \quad$ Surface integrals

$17.3 \quad$ Line integrals

$17.4 \quad$ Two-dimensional fields: line integrals in the plane and their relation to surface integrals on cylinders

$17.5 \quad$ Green's theorem

$17.6 \quad$ Divergence theorem

$17.7 \quad$ Stokes' theorem

$18 \quad$ Infinite series

$18.1 \quad$ Introduction and definitions

$18.2 \quad$ Tests for convergence of a series of constants

$18.3 \quad$ Power series expansions of functions

$18.4 \quad$ Taylor's theorem with remainder

$18.5 \quad$ Application to max-min theory for functions of two independent variables

$18.6 \quad$ Computations

$18.7 \quad$ Indeterminate forms

$18.8 \quad$ Fourier series

$18.9 \quad$ Convergence of power series: absolute convergence

$18.10 \quad$ Alternating series: conditional convergence

$19 \quad$ Complex numbers and functions

$19.1 \quad$ Invented number systems

$19.2 \quad$ The Argand diagram

$19.3 \quad$ The complex variable

$19.4 \quad$ Derivatives

$19.5 \quad$ Cauchy-Riemann differential equations

$19.6 \quad$ Complex series

$19.7 \quad$ Certain elementary functions

$19.8 \quad$ Logarithms

$20 \quad$ Differential equations

$20.1 \quad$ Introduction

$20.2 \quad$ Solutions

$20.3 \quad$ First-order equations with variables separable

$20.4 \quad$ First-order homogeneous equations

$20.5 \quad$ First-order linear equations

$20.6 \quad$ First-order equations with exact differentials

$20.7 \quad$ Special types of second-order equations

$20.8 \quad$ Linear equations with constant coefficients

$20.9 \quad$ Homogeneous linear second-order differential equations with constant coefficients

$20.10 \quad$ Nonhomogeneous linear second-order differental equations with constant coefficients

$20.11 \quad$ Higher-order linear differential equations with constant coefficients

$20.12 \quad$ Vibrations

$20.13 \quad$ Poisson probability distribution

Appendix $\text I \quad$ Determinants and linear equations

$\text A.1 \quad$ Indroduction

$\text A.2 \quad$ Determinants and linear equations

$\text A.3 \quad$ Determinants of order three

$\text A.4 \quad$ Determinants of order $n$

$\text A.5 \quad$ Properties of determinants of order $n$

$\text A.6 \quad$ Expansion by cofactors

$\text A.7 \quad$ Solution of simultaneous linear equations

$\text A.8 \quad$ Homogeneous linear equations

Appendix $\text {II} \quad$ Formulas from elementary mathematics

Appendix $\text {III} \quad$ Tables of functions

Answers to exercises

Index

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Further Editions

• 1951: George B. Thomas, Jr.: Calculus and Analytic Geometry
• 1996: George B. Thomas, Jr. and Ross L. Finney: Calculus and Analytic Geometry (9th ed.)

Source work progress

• 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous): Front endpapers: A Brief Table of Integrals: $69$.

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